scholarly journals Comparison of iterates of a class of differential operators in Roumieu spaces

2017 ◽  
Vol 101 (115) ◽  
pp. 261-266
Author(s):  
Rachid Chaili ◽  
Tayeb Mahrouz
Keyword(s):  
Differential Operators ◽  
Hypoelliptic Operators ◽  
Constant Coefficients

Considering a class of differential operators with constant coefficients including the hypoelliptic operators, we show that the comparison of the operators implies the inclusion between their spaces of Roumieu vectors.

Almost hypoelliptic differential operators with variable coefficients

1970 ◽  
Vol 67 (2) ◽  
pp. 287-293 ◽  
Author(s):  
Robert J. Elliott
Keyword(s):  
Differential Operators ◽  
A Priori ◽  
Adjoint Operator ◽  
Partial Differential Operator ◽  
Variable Coefficients ◽  
Hypoelliptic Operators ◽  
Standard Distribution ◽  
Strongly Regular ◽  
Constant Coefficients ◽  
Distribution Spaces

We have called a linear partial differential operator P almost hypoelliptic in some of the variables if the distribution u is strongly regular in those variables in every open subset where this is true of Pu. In an earlier paper (l) we have characterised almost hypoelliptic operators with constant coefficients. Following techniques of Hörmander (2) and Yoshikawa(5) we obtain in this paper some a priori inequalities for almost hypoelliptic operators and derive results about the solvability of the adjoint operator P*. Our notation for various standard distribution spaces and multi-indices follows Hörmander (2).

SIMILARITY OF OPERATORS ON TENSOR PRODUCTS OF SPACES AND MATRIX DIFFERENTIAL OPERATORS

2018 ◽  
Vol 106 (1) ◽  
pp. 19-30
Author(s):  
MICHAEL GIL’
Keyword(s):  
Differential Operators ◽  
Normal Operator ◽  
Variable Coefficients ◽  
Closed Convex Hull ◽  
Constant Matrix ◽  
Norm Estimates ◽  
Matrix Coefficients ◽  
Matrix Differential Operators ◽  
Constant Coefficients ◽  
Matrix Differential

Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

Boundary Value Problems for Harmonic Functions on the Heisenberg Group

1986 ◽  
Vol 38 (2) ◽  
pp. 478-512 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Heisenberg Group ◽  
Harmonic Functions ◽  
Poisson Kernel ◽  
Differential Operators ◽  
Special Functions ◽  
Continuous Functions ◽  
Group Representations ◽  
Hypoelliptic Operators ◽  
Analysis Group ◽  
Partial Differential Operators

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ, (γ ∊ C), associated to the Laplacian L0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ-harmonic). This is the topic of this paper.Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L0.

scholarly journals Lacunas for hyperbolic differential operators with constant coefficients I

1970 ◽  
Vol 124 (0) ◽  
pp. 109-189 ◽  
Author(s):  
M. F. Atiyah ◽  
R. Bott ◽  
L. Gårding
Keyword(s):  
Differential Operators ◽  
Constant Coefficients
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